SLIDE 1
Sets & Relations
Posets
SLIDE 2 Symmetric: Only self-loops & bidirectional edges Anti-symmetric: No bidirectional edges Transitive: Path from a to b implies edge (a,b)
has same last name as
≡
ancestor of
Landscape of Transitive Relations
Symmetric Anti-Symmetric
< ⊆ ≤
Acyclic Cannot follow a sequence
and get back to where you started from
!
SLIDE 3 ⊆ <
has same last name as
Landscape of Transitive Relations
Reflexive: All self-loops Irreflexive: No self-loops
≤
Symmetric Anti-Symmetric Irreflexive Reflexive
≡
Equivalences Partial Orders
Strict Partial Orders
Symmetric: Only self-loops & bidirectional edges Anti-symmetric: No bidirectional edges Transitive: Path from a to b implies edge (a,b)
ancestor of
SLIDE 4 Partial Order
A transitive, anti-symmetric and reflexive relation e.g. ≤ for integers, divides for integers, ⊆ for sets, “containment” for line-segments Equivalently, transitive and acyclic (and ir/reflexive) (a pair of bidirectional edges is a “cycle”) “Order” refers to these properties “Partial”: not every two elements need be “comparable” i.e., {a,b} s.t. neither a⊑b nor b⊑a e.g., neither A ⊆ B nor B ⊆ A
Strict partial order: irreflexive, rather than reflexive
SLIDE 5 Posets
Partially ordered set (a.k.a Poset) A non-empty set and a partial order over it Denoted like (S, ≼) e.g. S = {S1,S2,S3,S4,S5} where S1={0,1,2,3}, S2={1,2,3,4}, S3={1,2,3}, S4={3,4}, and S5 = {2}. Poset (S, ⊆) More generally, (S, ⊆) where S is any set of sets Verify: P⊆P; P⊆Q ⋀ Q⊆R → P⊆R; P⊆Q ⋀ Q⊆P → P=Q e.g. Divisibility poset: (Z+, | ) Verify: a|a ; a|b ⋀ b|c → a|c ; a|b ⋀ b|a → a=b
Check:
(no bidirectional edges),
- Transitive,
- Reflexive (all self-loops)
S1 S2 S3 S4 S5
SLIDE 6 Extremal & Extremum
Maximal & minimal elements of a poset (S, ≼) x∈S is maximal if ∄y∈S-{x} s.t. x≼y x∈S is minimal if ∄y∈S-{x} s.t. y≼x Need not exist (e.g., in (Z,≤)). Need not be unique when it exists (e.g., divisibility poset restricted to integers > 1) Claim: Every finite poset has at least one maximal and
Proof by induction on |S| [Exercise] x∈S is the greatest element if ∀y∈S, y≼x x∈S is the least element if ∀y∈S, x≼y
Useful in induction proofs about finite posets Need not exist. Unique when one exists.
SLIDE 7
Other Relations from a Poset
Consider partial order ≼ ≺ is the reflexive reduction of ≼ iff ≼ is the reflexive closure of ≺, and ≺ itself is irreflexive a≺b iff a≠b and a≼b ⊑ is the transitive reduction of ≼ iff ≼ is the transitive closure of ⊑, and ∀a,b ( a⊑b → ∄m∉{a,b}, a ≼ m ≼ b ) Well-defined for finite posets: Define a⊑b iff a≼b and ∄m∉{a,b}, a ≼ m ≼ b. [Prove by induction] Need not exist for infinite sets (e.g., for (R,≤), ⊑ defined as above is the equality relation)
SLIDE 8 Consider strict poset (Z+,⊏), where a ⊏ b iff b/a is prime Claim: | is the transitive closure of the reflexive closure
Claim: ⊏ is the transitive reduction of the reflexive reduction of | [Verify] Note: Divisibility poset has a transitive reduction even though it is infinite
Running Example
Divisibility poset: (Z+, | )
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SLIDE 9 For a poset (S, ≼), the transitive reduction of the reflexive reduction of ≼, if it exists, has all the information about the poset Recall: For finite posets, guaranteed to exist Hasse Diagram: the graph of this relation (with arrowheads implicit)
Hasse Diagram
16 8 12 4 6 9 10 14 15 2 3 5 7 11 13 1
SLIDE 10 Given a poset (S, ≼) and T ⊆ S Maximal element in T : x∈T s.t. ∀y∈T, x≼y → y=x Minimal element in T : x∈T s.t. ∀y∈T, y≼x → y=x Greatest element in T : x∈T s.t. ∀y∈T y≼x Least element in T : x∈T s.t. ∀y∈T, x≼y Upper Bound for T : x∈S s.t. ∀y∈T, y≼x Lower Bound for T : x∈S s.t. ∀y∈T, x≼y Least Upper Bound for T: Least in {x| x u.b. for T} Greatest Lower Bound for T: Greatest in {x| x l.b. for T}
Need not exist. Unique when one exists.
Bounding Elements
Need not exist. Need not be unique when one exists. Do exist in finite posets Need not exist. Unique when one exists.
SLIDE 11
Running Example
Lower bound When is c a lower bound for T={a,b}? c|a and c|b ⇒ c is a common divisor for {a,b} Greatest lower bound for {a,b} = gcd(a,b) Upper bound d is an upper bound for {a,b} ⇒ a|d, b|d ⇒ d a common multiple for {a,b} Least upper bound for {a,b} = lcm(a,b)
Divisibility poset: (Z+, | )
SLIDE 12
Total/Linear Order
In some posets every two elements are “comparable”: for {a,b}, either a⊑b or b⊑a Can arrange all the elements in a line, with all possible right-pointing edges (plus, all self-loops) If finite, has unique maximal and unique minimal elements (left and right ends)
SLIDE 13
Order Extension
A poset P’=(S,≤) is an extension of a poset P=(S,≼) if ∀a,b∈S, a ≼ b → a ≤ b Any finite poset can be extended to a total ordering (this is called topological sorting) Prove by induction on |S| Induction step: Remove a minimal element, extend to a total ordering, reintroduce the removed element as the minimum in the total ordering. For infinite posets? The “Order Extension Principle” is typically taken as an axiom! (Unless an even stronger axiom called the “Axiom of Choice” is used)
SLIDE 14 The totally ordered set (Z+, ≤ ), where ≤ is the standard “less-than-or-equals” relation, is an extension
Because a|b → a ≤ b Consider another totally ordered set (Z+, ⊑): For any (a,b) ∈ Z+ × Z+, a⊑b iff: a=1, or a,b both prime or both composite, and a ≤ b, or a prime and b composite (Z+, ⊑) extends the divisibility poset [Exercise]
Running Example
Divisibility poset: (Z+, | )
